Foundations — Trajectory optimization — minimum gravity loss, minimum drag loss

This page builds — from absolutely nothing — every letter, symbol, and idea the parent topic leans on. Read it top to bottom: each item uses only things defined above it.

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1. Speed $v$ and its change $\Delta v$

First, the plainest symbol of all: $v$ is the rocket's instantaneous speed — how fast it is moving right now, measured in metres per second ($\text{m/s}$). Picture the needle on a speedometer at one frozen instant.

The Greek letter $\Delta$ (delta) is shorthand for "the change in". So $\Delta v$ means "the change in speed $v$". If a rocket speeds up from $0$ to $3000\text{m/s}$, then $\Delta v=3000\text{m/s}$ — also in $\text{m/s}$.

Why does the topic need it? Because a rocket engine does not directly give you altitude or orbit — it gives you a speed budget. Every task and every loss is measured in the same unit, $\text{m/s}$, so they can be added and subtracted like money.

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2. Mass symbols: $m_0$, $m_f$, $m(t)$, and ${\dot{m}}_p$

A rocket gets lighter as it burns fuel — this is the whole reason rockets work. So we need names for its mass at different moments.

The little dot in ${\dot{m}}_p$ is Newton's notation for a rate of change per second. So ${\dot{m}}_p$ (read "m-dot-p") is how many kilograms of fuel leave the nozzle each second.

Read the figure below so these symbols stop being abstract. The purple curve is $m(t)$ sliding downhill as time runs. Notice three things the figure is pointing at: the mint dashed line at the top is $m_0$ (the full mass we start with); the coral dashed line at the bottom is $m_f$ (the empty shell we end with); and the steepness of the purple slope is exactly ${\dot{m}}_p$ — the steeper it drops, the faster fuel is leaving. The shaded band between the curve and $m_f$ is the propellant $m_p$ being spent.

Why does the topic need these? The Tsiolkovsky equation compares $m_0$ to $m_f$: a rocket's speed budget depends on the ratio of full-to-empty mass, not on either alone.

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3. The integral sign $\int$ — adding up a changing thing

Before we can build the speed budget or any loss, we need one piece of notation that appears everywhere in this topic: the integral sign $\int$.

We will use it in two forms: summing over mass (as fuel drains from $m_0$ to $m_f$, next section) and summing over time (from launch to burnout, §8). With this one tool in hand, the derivations below stay honest — every $\int$ you meet is now defined.

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4. Exhaust speed $v_e$, the natural logarithm $\ln$, and where Tsiolkovsky comes from

Now a new tool appears: $\ln$, the natural logarithm.

The word ideal flags that this is the perfect speed budget — with no gravity, no air, no wasted pointing. Reality subtracts from it; those subtractions are the "losses".

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5. Local gravity $g$ (and the honest truth about $g(t)$)

Gravity weakens with altitude, so when we want to be honest over a long climb we write $g(t)$ — gravity as a function of time.

Why the topic needs it: gravity is the force the rocket fights while climbing, and it is the free force that gently turns the rocket over later. Both roles are measured through $g$. Exactly how gravity gets split into a speed-stealing part and a turning part is the job of the flight-path angle (§6) and the sine/cosine split (§7) — which is also where we will finally plug real numbers into a gravity-loss integral.

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6. The flight-path angle $\gamma$ — the star of the show

This is the single most important symbol in the whole topic. $\gamma$ (Greek "gamma") is the angle between the rocket's velocity and the local horizon (the flat ground direction).

Walk through the figure below. The horizontal grey arrow is the local horizon — the "flat ground" direction. The coral arrow is the rocket's velocity. The purple wedge between them, marked $\gamma$, is the flight-path angle: as it opens toward $90^{\circ }$ the coral arrow swings toward straight up; as it closes toward $0^{\circ }$ the coral arrow lies flat. The mint arrow shows gravity, always pointing straight down regardless of $\gamma$ — that fixed downward direction is exactly why $\gamma$ alone controls how gravity relates to the motion.

That is why every loss formula in this topic contains $\gamma$. See also Flight-path angle and equations of motion.

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7. Sine and cosine — WHY they appear, on the triangle

Gravity pulls straight down, but the rocket moves along its velocity at angle $\gamma$. To ask "how much of gravity fights the motion?" we must split the downward pull into two pieces: one along the velocity, one perpendicular to it. Splitting a straight-down arrow using an angle is exactly the job sine and cosine were invented for.

Study the figure below carefully — it is the mechanical heart of the whole topic. The mint arrow is the full gravity $g$ pointing straight down. We break it into two dashed pieces. The coral dashed arrow, laid along the velocity direction, is $g\sin \gamma$ — the part that pulls backward against the motion and steals speed. The lavender dashed arrow, perpendicular to the velocity, is $g\cos \gamma$ — the part that only curves the path without slowing it. Watch the extremes:

• $g\sin \gamma$ (speed-stealer): straight up ($\gamma =90^{\circ }$), $\sin 90^{\circ }=1$, so all of $g$ fights you; flat ($\gamma =0^{\circ }$), $\sin 0^{\circ }=0$, so none fights your speed.
• $g\cos \gamma$ (path-curver): straight up, $\cos 90^{\circ }=0$, no turning; flat, $\cos 0^{\circ }=1$, maximum turning.

This split gives us $g\sin \gamma$ (→ gravity loss) and $g\cos \gamma$ (→ the gravity-turn steering). Both come straight off this one figure.

Now that $\gamma$, sine, and the integral are all defined, we can finally read a gravity-loss integral with real numbers:

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8. Burn time $t_b$ and time-integrals over the burn

The losses are written as ${\int }_0^{t_b}(\dots )dt$ — the same integral tool from §3, now summing over time.

$t_b$ (subscript $b$ = "burnout") is simply the clock time when the engine cuts off — the upper edge of every loss sum. It is set by the Thrust-to-weight ratio and burn time. Because $\gamma$, $g$, $v$, and $m(t)$ all drift during those seconds, only an integral (not plain multiplication) gives the true loss.

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9. Force symbols: thrust $T$ and drag $D$

Unpacking that formula, symbol by symbol:

• $\rho$ (Greek "rho") — air density, kilograms of air per cubic metre. Thick near the ground, near-zero in space. Picture how much "stuff" the rocket must shove aside.
• $v$ — the rocket's speed through the air (the same $v$ from §1).
• $v^2$ — speed squared: doubling speed quadruples drag. This squared term is why staying fast in thick air is so punishing.
• $C_D$ — the drag coefficient, a shape number ($\sim 0.3$ for a sleek rocket): how streamlined the body is.
• $A$ — the frontal area, the cross-section the rocket presents to the wind, in ${\text{m}}^2$.

Why the topic needs these: the drag-loss integrand is $D/m$, so every one of these numbers feeds the loss you are trying to minimize. The dangerous peak of $\rho v^2$ is called max-Q — see Aerodynamic Drag and Max-Q.

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10. Naming each loss, then putting them together

Before the formulas, here is a one-line meaning for every $\Delta v$ term you will meet — each is a chunk of speed, in $\text{m/s}$:

The angle $\alpha$ used by the steering term needs its own picture-anchored definition:

Now every symbol is defined, so the loss formulas read cleanly (note the bare $m$ inside them means the instantaneous $m(t)$ from §2):

The grand bookkeeping is then: $\Delta v_{\text{ideal}}=\Delta v_{\text{orbit}}+\Delta v_{\text{grav}}+\Delta v_{\text{drag}}+\Delta v_{\text{steer}}$ where $\Delta v_{\text{orbit}}$ is set by Orbital insertion and required orbital velocity.

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11. How these foundations feed the topic

How to read this map: each box is a foundation from above, and an arrow means "feeds into". Follow the flow from the scattered starting symbols (top) down to the single goal (bottom): mass, exhaust speed and $\ln$ combine into the ideal budget; gravity and the angle $\gamma$ get split by sine/cosine into the speed-stealing and path-curving parts; air density and speed build the drag force; and the integral wraps the gravity and drag pieces into losses. Everything then converges on the one node "minimize total loss".

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Equipment checklist

Read each question, answer aloud, then reveal. If any stumps you, reread its section above.